Section 2.3

Measures of Central Tendency

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Section 2.3

Section 2.3 Objectives

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Determine the mean, median, and mode of a population and of a sample

Determine the weighted mean of a data set and the mean of a frequency distribution

Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each

Notations

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denotes the addition of a set of values

x is the variable usually used to represent the individual data values

n represents the number of data values in a sample

N represents the number of data values in a population

Measures of Central Tendency

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Measure of central tendencyA value that represents a typical, or

central, entry of a data set.Most common measures of central

tendency:MeanMedianModeMidrange

Measure of Central Tendency: Mean

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Mean (average)The sum of all the data entries divided by

the number of entries.Sigma notation: Σx = add all of the data

entries (x) in the data set.Population mean:

Sample mean:

x

N

xx

n

Example: Finding a Sample Mean

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The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?

872 432 397 427 388 782 397

Solution: Finding a Sample Mean

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872 432 397 427 388 782 397

• The sum of the flight prices is

Σx = 872 + 432 + 397 + 427 + 388 + 782 + 397 = 3695

• To find the mean price, divide the sum of the prices by the number of prices in the sample

3695527.9

7

xx

n

The mean price of the flights is about $527.90.

Measure of Central Tendency: Median

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MedianThe value that lies in the middle of the data

when the data set is ordered.Measures the center of an ordered data set by

dividing it into two equal parts.If the data set has an

odd number of entries: median is the middle data entry.

even number of entries: median is the mean of the two middle data entries.

Example: Finding the Median

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The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices.

872 432 397 427 388 782 397

Solution: Finding the Median

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872 432 397 427 388 782 397

• First order the data.

388 397 397 427 432 782 872

• There are seven entries (an odd number), the median is the middle, or fourth, data entry.

The median price of the flights is $427.

Example: Finding the Median

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The flight priced at $432 is no longer available. What is the median price of the remaining flights?

872 397 427 388 782 397

Solution: Finding the Median

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872 397 427 388 782 397

• First order the data.

388 397 397 427 782 872

• There are six entries (an even number), the median is the mean of the two middle entries.

The median price of the flights is $412.

397 427Median 412

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Measure of Central Tendency: Mode

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ModeThe data entry that occurs with the greatest

frequency.If no entry is repeated the data set has no

mode.If two entries occur with the same greatest

frequency, each entry is a mode (bimodal).

Example: Finding the Mode

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The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices.

872 432 397 427 388 782 397

Solution: Finding the Mode

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872 432 397 427 388 782 397

• Ordering the data helps to find the mode.

388 397 397 427 432 782 872

• The entry of 397 occurs twice, whereas the other data entries occur only once.

The mode of the flight prices is $397.

Example: Finding the Mode

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At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses?

Political Party Frequency, f

Democrat 34

Republican 56

Other 21

Did not respond

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Solution: Finding the Mode

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Political Party Frequency, f

Democrat 34

Republican 56

Other 21

Did not respond

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The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation.

Measure of Central Tendency: Midrange

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MidrangeMidrange is the value midway between

the highest andlowest values in the original data set.

Midrange =

(highest value + lowest value)/2

Example: Finding the Midrange

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The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the Midrange of the flight prices.

872 432 397 427 388 782 397

Midrange = (388+872)/2 = 630

The midrange of the flight prices is $630.

Comparing the Mean, Median, Mode, and Midrange

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All four measures describe a typical entry of a data set.

Advantage of using the mean:The mean is a reliable measure because it

takes into account every entry of a data set.Disadvantage of using the mean:

Greatly affected by outliers (a data entry that is far removed from the other entries in the data set).

Example: Comparing the Mean, Median, Mode, and Midrange

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Find the mean, median, mode, and midrange of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers?

Ages in a class

20 20 20 20 20 20 21

21 21 21 22 22 22 23

23 23 23 24 24 65

Solution: Comparing the Mean, Median, and Mode

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Mean:20 20 ... 24 65

23.8 years20

xx

n

Median:21 22

21.5 years2

20 years (the entry occurring with thegreatest frequency)

Ages in a class

20 20 20 20 20 20 21

21 21 21 22 22 22 23

23 23 23 24 24 65

Mode:

Midrange:

(20+65)/2 = 42.5 years


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Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years Midrange = 42.5 years• The mean takes every entry into account,

but is influenced by the outlier of 65. • The median also takes every entry into

account, and it is not affected by the outlier.

• In this case the mode exists, but it doesn't appear to represent a typical entry.

• The midrange depends only on the lowest and highest values. So, it does not represent a typical data entry.


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Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set.

In this case, it appears that the median best describes the data set.

Weighted Mean

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Weighted Mean

The mean of a data set whose entries have varying weights.

, where w is the weight of each entry x

( )x wx

w

Example: Finding a Weighted Mean

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You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A?

Solution: Finding a Weighted Mean

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Source Score, x Weight, w

x∙w

Test Mean 86 0.50 86(0.50)= 43.0

Midterm 96 0.15 96(0.15) = 14.4

Final Exam 82 0.20 82(0.20) = 16.4

Computer Lab

98 0.10 98(0.10) = 9.8

Homework 100 0.05 100(0.05) = 5.0

Σw = 1 Σ(x∙w) = 88.6

( ) 88.688.6

1

x wx

w

Your weighted mean for the course is 88.6. You did not get an A.

Mean of Grouped Data

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Mean of a Frequency DistributionApproximated by

where x and f are the midpoints and frequencies of a class, respectively

( )x fx n f

n

Finding the Mean of a Frequency Distribution

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In Words In Symbols

( )x fx

n

(lower limit)+(upper limit)

2x

( )x f

n f

1. Find the midpoint of each class.

2. Find the sum of the products of the midpoints and the frequencies.

3. Find the sum of the frequencies.

4. Find the mean of the frequency distribution.

Example: Find the Mean of a Frequency Distribution

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Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session.

Class Midpoint Frequency, f

7 – 18 12.5 6

19 – 30 24.5 10

31 – 42 36.5 13

43 – 54 48.5 8

55 – 66 60.5 5

67 – 78 72.5 6

79 – 90 84.5 2

Solution: Find the Mean of a Frequency Distribution

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Class Midpoint, x

Frequency, f

(x∙f)

7 – 18

12.5 6 12.5∙6 = 75.0

19 – 30

24.5 10 24.5∙10 = 245.0

31 – 42

36.5 13 36.5∙13 = 474.5

43 – 54

48.5 8 48.5∙8 = 388.0

55 – 66

60.5 5 60.5∙5 = 302.5

67 – 78

72.5 6 72.5∙6 = 435.0

79 – 90

84.5 2 84.5∙2 = 169.0

n = 50 Σ(x∙f) = 2089.0

( ) 208941.8 minutes

50

x fx

n

The Shape of Distributions

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Symmetric Distribution

• A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.

• Data is symmetric if the left half of its histogram is roughly a mirror of its right half.


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Uniform Distribution (rectangular)

• All entries or classes in the distribution have equal or approximately equal frequencies.

• Symmetric.


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Skewed Left Distribution (negatively skewed)

• The “tail” of the graph elongates more to the left.

• The mean is to the left of the median.

• mean < median < mode


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Skewed Right Distribution (positively skewed)

• The “tail” of the graph elongates more to the right.

• The mean is to the right of the median.

• mean > median > mode

Skewness

Mode = Mean = Median

SKEWED LEFT

(negatively)

SYMMETRIC

Mean Mode Median

SKEWED RIGHT(positively)

Mean Mode Median

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Practice Questions

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Q. (2.9)Given the following data set

61, 11, 1, 3, 2, 30, 18, 3, 7

Find the mean, median, mode and midrange. Which measure of

the center of data might be the best in this case?

Practice Questions

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Q. (2.10)The cost per load (in cents) of 35 laundry

detergents tested by a consumer organization is shown below.

Find the mean of the costs.

262-68055-61148-54641-47534-40

1227-33720-26213-19

FrequencyClass Limits

Section 2.3 Summary

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Determined the mean, median, and mode of a population and of a sample

Determined the weighted mean of a data set and the mean of a frequency distribution

Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each

Documents

Section 2.3